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Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method

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1 Author(s)
Vershynin, R. ; Dept. of Math., California Univ., Davis, CA

Spielman and Teng proved that the shadow-vertex simplex method had polynomial smoothed complexity. On a slight random perturbation of arbitrary linear program, the simplex method finds the solution after a walk on the feasible polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/sigma. We show that the length of walk is actually polylogarithmic in the number of constraints n. We thus improve Spielman-Teng's bound on the walk O*(n86d 55sigma-30) to O(max(d5log2n, d9 log4d, d 3sigma-4)). This in particular shows that the tight Hirsch conjecture n - d on the diameter of polytopes is not a limitation for the smoothed linear programming. Random perturbations create short paths between vertices. We propose a randomized phase-I for solving arbitrary linear programs. Instead of finding a vertex of a feasible set, we add a vertex at random to the feasible set. This does not affect the solution of the linear program with constant probability. So, in expectation it takes a constant number of independent trials until a correct solution is found. This overcomes one of the major difficulties of smoothed analysis of the simplex method - one can now statistically decouple the walk from the smoothed linear program. This yields a much better reduction of the smoothed complexity to a geometric quantity - the size of planar sections of random polytopes. We also improve upon the known estimates for that size

Published in:

Foundations of Computer Science, 2006. FOCS '06. 47th Annual IEEE Symposium on

Date of Conference:

Oct. 2006